A general Riemann complete integral in the plane (Q1187276)

This interesting paper proves the following Cousin type lemma: Given a closed interval \(A\) in \(\mathbb{R}^ 2\), and a positive function (gauge) \(\delta\) on \(A\), there exists a \(10^{-3}\)-regular, \(\delta\)-fine and special partition of \(A\). Recall that a partition \(P\) of \(A\) is a finite collection \(P=\{(A_ 1,x_ 1),\dots,(A_ k,x_ k)\}\), where the mutually non-overlapping closed intervals \(A_ j\) have union \(A\) and \(x_ j\in A_ j\) \((1\leq j\leq k)\), the regularity \(r(A_ j)\) of \(A_ j\) is the ratio of the length of the smaller side of \(A_ j\) and the length of the larger side of \(A_ j\) and the regularity \(r(P)\) of \(P\) is the smallest of the \(r(A_ j)\). Then \(P\) is said to be \(r\)-regular if \(r(P)\geq r\). Finally, \(P\) is called special if each \(x_ j\) belongs to a vertex of \(A_ j\) and \(\delta\)-fine if each \(A_ j\) is contained in the ball \(B[x_ j,\delta(x_ j)]\). This Cousin's lemma allows to define, in the line of previous work of Kurzweil, Henstock, Mawhin and Pfeffer, a Riemann-type integral over \(A\) which integrates the divergence of simply differentiable vector fields. It remains an interesting open problem to know if the above lemma is valid in higher dimensions.